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We prove that, if a discrete group $G$ is not inner amenable, then the unitgroup of the ring of operators affiliated with the group von Neumann algebra of$G$ is non-amenable with respect to the topology generated by its rank metric.This provides examples of non-discrete irreducible, continuous rings (in vonNeumann's sense) whose unit groups are non-amenable with regard to the ranktopology. Our argument establishes and uses connections with Eymard--Greenleafamenability of the action of the unitary group of a $\mathrm{II}_{1}$ factor onthe associated space of projections of a fixed trace.

Group von Neumann algebras, inner amenability, and unit groups of continuous rings